# Homework Solution - Set 1 ```CSE 396
Fall2008
Introduction to the Theory of Computation
Homework Solution - Set 1
Due: Friday 09/05/08
1. Textbook, Page 26, Exercise 0.7.
Let A = {a, b, c}. Let Z+ be the set of positive integers.
a. The following relation R1 is a binary relation on A that is
reflexive and symmetric but not transitive:
R1 = {(a, a), (b, b), (c, c), (a, b), (b, a), (b, c), (c, b)}.
The following relation R2 is an infinite such relation, which
is defined on Z+ :
R2 = {(a, b) | ka − bk ≤ 1}
b. The following relation R3 is a binary relation on A that is
reflexive and transitive but not symmetric:
R3 = {(a, a), (b, b), (c, c), (a, b), (b, c), (a, c)}.
The normal ≤ relation defined on Z+ is an infinite such
relation.
c. The following relation R4 is a binary relation on A that is
symmetric and transitive but not reflexive:
R4 = {(a, a), (b, b), (a, b), (b, a)}.
The following relation R5 is an infinite such relation, which
is defined on Z+ :
R2 = {(a, b) | both a and b are even. }
Note that the key point in constructing a relation that is
symmetric and transitive but not reflexive is that there has
to be some “isolated” element in the base set that is not
related to any other element.
1
2. Textbook, Page 27, Exercise 0.8.
The graph is as follows:
1
4
2
3
The thickened lines indicate a path from node 3 to node 4. The
degree of node 1 is 3 and that of node 3 is 2.
2
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